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Squeezing in the multiphoton Jaynes-Cummings model
Volume 124, number 6,7
5 October 1987
PHYSICS LETTERS A
SQUEEZING IN THE MULTIPHOTON JAYNES-CUMMINGS MODEL A.S. SHUMOVSKY, FAM LE KIEN and E.I. ALISKENDEROV Laboratory of Theoretical Physics, Joint Institutefor Nuclear Research, Head Post Ofice P.O. Box 79, Moscow 101000, USSR
Received 6 May 1987; revised manuscript received 24 August 1987; accepted for publication 26 August 1987 Communicated by B. Fricke
Squeezing in the exactly solvable multiphoton Jaynes-Cummings
A substantial interest has centered upon the recent experimental observations [ l-3 ] of squeezed field states [ 41 which present a new nonclassical effect in radiation theory and may have potential application in optical communication and gravitational wave detection. Several schemes for producing squeezed states have been analysed. Among them are parametric amplifiers [ 5,3], four-wave mixing [ 6,1,2], two-photon interaction with an absorber [ 7-91, twophoton lasers [ lo- 121, resonance fluorescence [ 13,14 1, cooperative Dicke systems and others. On the other hand, the progress in the realization of a single Rydberg atom in a resonant cavity [ 151 and the first observation of quantum collapse and revival in a one-atom maser [ 161 make now possible testing the simplest quantum electrodynamic models of oneatom one-mode systems [ 171. It has been shown that light squeezing is possible in the Jaynes-Cummings model with a coherent cavity field [ 181 and in the Jaynes-Cummings-type models with special baretype initial states [ 19,2 11. the magnitudes of squeezing in these systems are however rather small ( < 20% [ 181, ~25% [21] and ~42% [ 191). Moreover, the squeezing obtained numerically in ref. [ 181 appears not at once for t > 0, but only after some finite interval of interaction time, and the initial conditions for squeezing in refs. [ 19,211 are too specific and obviously only of academic interest. The aim of the present paper is to report the results showing that states containing a large amount of squeezing can be obtained from the exactly solvable multiphoton Jaynes-Cummings model with a coherent cavity field.
model is examined.
We consider a two-level atom interacting with a single-mode radiation field in a lossless resonant cavity via the m-photon-transition mechanism. The effective hamiltonian for this system in the rotating wave approximation is H=Awa+a+AooR’+hg(R+a”+R-a+“),
(1)
where o and w. = mw are the frequencies of the field and the atom, respectively, g is the multiphoton atom-radiation coupling constant, m is the photon multiple, R’, R + and R - are the atomic pseudospin operators, and a+ and a are the creation and annihilation operators of the field. We denote by ( + ), I- ) the excited and ground states of the atom and by ( n) the Fock states of the field. For the atom initially in the ground state ( - ) and the field initially in a coherent 1z), Iz>=exp(-t]z]*)
(2)
g Eln>,
.=0&i
the wavefunction of the total system in the interaction picture is found from the hamiltonian (1) to be Iw(t))=~~ol-;n)A’“)(t)+l+;n-m)Al”)(t)] xexp( -f ]zJ’)z”lJn! ,
(3)
where A!!qt)=cos[gt&qCF)!])
A’,“‘(t)
= -i sin[gtJn!lo!]
(da) .
(4b)
351
Volume
124, number
6,7
PHYSICS
Hence, the mean photon number (~+a), the mean photon amplitude (a) and the mean square photon amplitude (a’) are easily derived to read
LETTERS
s
= a
A
5 October
(A&d2- (Au,):,, (Au&,
=4((a,-(u,))2)-l. (u+u)=a~=n--m
T P,JA’,“‘)2, ll=,,?
f P,A !“)*A’;‘+‘),), ,I=1>1
n=
+2 Re(u2e2i(wr-s))
-4(Im(uei(w’-@)))2.
$,++A9?+2)
111
x&l-ml(n+2)][1-ml(n+l)]
. >
(10)
It is seen from eqs. ( 5) and (10) that the quantities oo, (T, and rs2 are real numbers, and therefore, the optimum choice of 8 for squeezing should be 8 = p, where p is the phase of z, i.e. z= fi”2exp(iq). Then, eqs. (10) become
F PnAL”‘*A’“+z) eZiwt(u2) =z2g2 =z2 n=O + fPnA
(9)
In terms of the photon operators, we readily find that S, =2(a+u)
+
1987
s, =2l70+2ficsz -4na: (5)
)
(1 la)
s, = 2ao - 2Aa2 .
(1 lb)
Here P, is the poissonian distribution corresponding to the coherent initial state (2) of the field
For very short times (gtc 1 ), we find from eqs. (1 la), (5) and (4) the asymptotic expressions
P, =ePfifi”/n!
s, N - fn(gt)4 )
(6)
and fi= (z (’ is the dimensionless intensity of the field. It should be mentioned here that the time behaviour of the atomic inversion and photon number has been investigated in ref. [ 221. The appearance of the so-called Cummings collapse, revivals and recorrelations has been shown for this multiphoton model. We introduce the two slowly varying hermitian quadrature components a I, u2 of the field, defined by a, =j[ue
i(wr-0)
_tu
u2=(1/2i)[aei(“‘-e)_u+e-i(wl-s)],
+e
-i(wr-0)
19
(7)
where 19is a phase angle that may be chosen at will. The commutation of a, and a2 is [u,, uz] = hi. The variances (Au,)~E (a f) - (u,)~ (i= 1, 2) satisfy the uncertainty relation (Au,)~(Au~)~> +e A state of the field is said to be squeezed when one of the quadrature components a, and a2 satisfies the relation (AU,)2<(AUi)$,h=i [18]. The condition for squeezing in the quadrature component can simply be written as S,
(8)
= -m(m-1)A+‘(gt)2,
if
m=l,
if
m>2,
(12)
These negative expressions indicate the immediate appearance of sueezing in a, for any photon multiple m and arbitrary nonzero intensity A after switching on the atom-field interaction. Such a behaviour is absent in the case when the atom is initially in the excited state [ 181. Analogously, the asymptotic expressions of S2 at very short times (gt << 1) are found from eqs. (11 b), (5) and (4) to be s, N fn(gt)4 , =m(m-
1)A*-‘(gt)2
,
if
m=l,
if
ma2.
(13)
These positive expressions indicatre the lack of squeezing in a2 at the beginning of interaction for any photon multiple m and any initial field intensity n. It should be noted that for the particular case m = 1 the first equation in (13) and the fact that squeezing occurs in a, at the onset of interaction, are in agreement with the results obtained recently by Butler and Drummond [ 201 for a cooperative Dicke system.
5 October 1987
PHYSICS LETTERS A
Volume 124, number 6,7
Fig. 1 presents the time evolution of S, computed numerically from eqs. (1 ), (5) and (4) for various intensities A of the coherent initial field and various photon multiples m. As soon as t>O, we observe nonclassical negative values of S,. For the cases (m=l,fi=0.2), (m=2,A=l.l2)and(m=3,A=3) the maximum magnitudes of squeezing in the region of short times are S, x - 0.28, - 0.49 and - 0.57 (i.e. 28%, 49% and 57%), respectively. As time goes on, S, starts oscillating and then reaches positive values. The long-time behaviour of S, is characterized by recoveries of squeezing. The squeezing in a, appears,
disappears, and later may appear again, see fig. Id. The maximum magnitude of squeezing recovered ~~1.12, m=2, see again (e.g., x52%forgt=l7.28, fig. 1d) may be larger than the maximum magnitude of squeezing in the short-time region ( ~49% for gtx0.92). It is seen from fig. 1 that for the larger intensity fi the duration of the first squeezing is generally shorter (except for the cases when overlapping of the first and second squeezing regions occurs). In figs. 2a and 2b, we plot S, versus time gt for the cases (m=l, E~0.2) and (m=2, E=5). It is clear from fig. 2 that squeezing may also occur in the field
0.1
0 cn’ -0.1
- 0.2 m=2
-0.6 0
0.4
-0.3 0
2
1
1.2
1.6
2.0
12
16
20
gt -
5
3
0.8
gt -
-0.6 _ 0
CC) _ m=3 0.2
0.L
0.6 qt -
0.8
1.0
1.2
0
4
8 gt -
Fig. 1. Time evolution of S,. (a) m = 1, short times: gf < 5. Here, the maximum magnitude S , z -0.28 of the first squeezing occurs for ti% 0.2 at gt = 2.75. (b) m = 2, short times gt < 2. The maximum magnitude 5’, z - 0.49 of the first squeezing occurs for ffz 1.12at gt= 0.92. (c) m= 3, short times gtc 1.2. The maximum magnitude S ,x -0.57 of the first squeezing occurs for ii=3 at gtz0.27. (d) Long times: gl< 20. The full line corresponds to the case m = 1,A= 0.2. The dashed line corresponds to the case m = 2, A= 1.12.The large magnitudes form=1.fi=0.2,gr~l965and.S of squeezing: S ,z-0.31 ,~-0.52form=2,ri=1.12,gt~17.28areobtained.
353
Volume
124, number
6.7
PHYSICS
(a) i
0.6 0.5 0.4
LETTERS
A
5 October
1987
able choice of the field intensity, the large magnitudes of squeezing have been obtained. It is however very difficult to detect this transient squeezing effect in an experiment. In recent experiments with Rydberg atoms in high-Q cavities it was possible to measure only the inversion of the atom but not the field itself. The connection between the evolution of the squeezing and that of the atomic inversion will be the subject of a subsequent paper.
References
[ I] R.E. Slusher, L.W. Hollberg, B. Yurke, J.C. Mertz and J.F. - 0.3 I 0
4
8
12
J 20
16 qt -
2.9
3.0
3.1
3.2
3.3
gt Fig. 2. Time evolution of&. (a) M= 1, A=0.2, long times: gt<20. (b) m=2, ri=S, the time interval 2.9cgfc3.3 for the first squeezing in uz.
component a2. The delay of squeezing in a2 for the cases examined here is seen. To conclude briefly, we have obtained squeezed states in the exactly solvable multiphoton Jaynes-Cummings model, where the atom is initially in the ground state and the fields is in a coherent state. The immediate appearance of squeezing in a, for short times of interaction has been established. Due to the photon multiplicity and the suit-
354
Valley, Phys. Rev. Lett. 55 (1985) 2409. [2] R.M. Shelby, M.D. Levenson, S.H. Perlmutter, R.G. Devoe and D.F. Walls, Phys. Rev. Lett. 57 (1986) 691. [ 31 Ling An Wu, H.J. Kimble, J.L. Hall and Huifa Wu, Phys. Rev. Lett. 57 (1986) 2520. [4] D.W. Walls, Nature 306 (1983) 141. [ 51 M.J. Collett and D.F. Walls, Phys. Rev. A 32 (1985) 2887. [6] J.R. Klauder, S.L. McCall and B. Yurke, Phys. Rev. A 33 (I 986) 3204. [ 71 L. Mandel, Opt. Commun. 42 (1982) 437. [8] M.H. Hillery, MS. Zubairy and K. Wodkiewicz, Phys. Lett. A 103 (1984) 259. [9] R. Loudon, Opt. Commun. 49 (1984) 67. [IO] H.P. Yuen, Phys. Rev. A 13 (1976) 2226. [ 1 I ] L.A. Lugiato and G. Strini, Opt. Commun. 41 (1982) 374. [ 121 M.D. Reid and D.F. Walls, Phys. Rev. A 28 (1983) 332. [ 131 D.F. Walls and P. Zoller, Phys. Rev. Lett. 47 (198 I ) 709. [ 141 P.A. Lakshmi and G.S. Agarwall, Phys. Rev. A 29 (1984) 2260. [ 151 S. Haroche and J.M. Raimond, in: Advances in atomic and molecular physics, Vol. 20, eds. D.R. Bates and B. Bederson (Academic Press, New York, 1985) p. 347. [ 161 G. Rempe, H. Walther and N. Klein, Phys. Rev. Lett. 58 (1987) 353. [ 171 H.I. Yoo and J.H. Eberly, Phys. Rep. 118 (1985) 239. [ IS] P. Meystre and M.S. Zubairy, Phys. Lett. A 89 (1982) 390. [ 191 G. Compagno, J.S. Peng and F. Persico, Opt. Commun. 57 (1986) 415. [20] M. Butler and P.D. Drummond, Opt. Acta 33 (1986) 1. [21] P.L. Knight, Phys. Ser. T 12 (1986) 51. [ 221 S. Singh, Phys. Rev. A 25 (1982) 3206.
5 October 1987
PHYSICS LETTERS A
SQUEEZING IN THE MULTIPHOTON JAYNES-CUMMINGS MODEL A.S. SHUMOVSKY, FAM LE KIEN and E.I. ALISKENDEROV Laboratory of Theoretical Physics, Joint Institutefor Nuclear Research, Head Post Ofice P.O. Box 79, Moscow 101000, USSR
Received 6 May 1987; revised manuscript received 24 August 1987; accepted for publication 26 August 1987 Communicated by B. Fricke
Squeezing in the exactly solvable multiphoton Jaynes-Cummings
A substantial interest has centered upon the recent experimental observations [ l-3 ] of squeezed field states [ 41 which present a new nonclassical effect in radiation theory and may have potential application in optical communication and gravitational wave detection. Several schemes for producing squeezed states have been analysed. Among them are parametric amplifiers [ 5,3], four-wave mixing [ 6,1,2], two-photon interaction with an absorber [ 7-91, twophoton lasers [ lo- 121, resonance fluorescence [ 13,14 1, cooperative Dicke systems and others. On the other hand, the progress in the realization of a single Rydberg atom in a resonant cavity [ 151 and the first observation of quantum collapse and revival in a one-atom maser [ 161 make now possible testing the simplest quantum electrodynamic models of oneatom one-mode systems [ 171. It has been shown that light squeezing is possible in the Jaynes-Cummings model with a coherent cavity field [ 181 and in the Jaynes-Cummings-type models with special baretype initial states [ 19,2 11. the magnitudes of squeezing in these systems are however rather small ( < 20% [ 181, ~25% [21] and ~42% [ 191). Moreover, the squeezing obtained numerically in ref. [ 181 appears not at once for t > 0, but only after some finite interval of interaction time, and the initial conditions for squeezing in refs. [ 19,211 are too specific and obviously only of academic interest. The aim of the present paper is to report the results showing that states containing a large amount of squeezing can be obtained from the exactly solvable multiphoton Jaynes-Cummings model with a coherent cavity field.
model is examined.
We consider a two-level atom interacting with a single-mode radiation field in a lossless resonant cavity via the m-photon-transition mechanism. The effective hamiltonian for this system in the rotating wave approximation is H=Awa+a+AooR’+hg(R+a”+R-a+“),
(1)
where o and w. = mw are the frequencies of the field and the atom, respectively, g is the multiphoton atom-radiation coupling constant, m is the photon multiple, R’, R + and R - are the atomic pseudospin operators, and a+ and a are the creation and annihilation operators of the field. We denote by ( + ), I- ) the excited and ground states of the atom and by ( n) the Fock states of the field. For the atom initially in the ground state ( - ) and the field initially in a coherent 1z), Iz>=exp(-t]z]*)
(2)
g Eln>,
.=0&i
the wavefunction of the total system in the interaction picture is found from the hamiltonian (1) to be Iw(t))=~~ol-;n)A’“)(t)+l+;n-m)Al”)(t)] xexp( -f ]zJ’)z”lJn! ,
(3)
where A!!qt)=cos[gt&qCF)!])
A’,“‘(t)
= -i sin[gtJn!lo!]
(da) .
(4b)
351
Volume
124, number
6,7
PHYSICS
Hence, the mean photon number (~+a), the mean photon amplitude (a) and the mean square photon amplitude (a’) are easily derived to read
LETTERS
s
= a
A
5 October
(A&d2- (Au,):,, (Au&,
=4((a,-(u,))2)-l. (u+u)=a~=n--m
T P,JA’,“‘)2, ll=,,?
f P,A !“)*A’;‘+‘),), ,I=1>1
n=
+2 Re(u2e2i(wr-s))
-4(Im(uei(w’-@)))2.
$,++A9?+2)
111
x&l-ml(n+2)][1-ml(n+l)]
. >
(10)
It is seen from eqs. ( 5) and (10) that the quantities oo, (T, and rs2 are real numbers, and therefore, the optimum choice of 8 for squeezing should be 8 = p, where p is the phase of z, i.e. z= fi”2exp(iq). Then, eqs. (10) become
F PnAL”‘*A’“+z) eZiwt(u2) =z2g2 =z2 n=O + fPnA
(9)
In terms of the photon operators, we readily find that S, =2(a+u)
+
1987
s, =2l70+2ficsz -4na: (5)
)
(1 la)
s, = 2ao - 2Aa2 .
(1 lb)
Here P, is the poissonian distribution corresponding to the coherent initial state (2) of the field
For very short times (gtc 1 ), we find from eqs. (1 la), (5) and (4) the asymptotic expressions
P, =ePfifi”/n!
s, N - fn(gt)4 )
(6)
and fi= (z (’ is the dimensionless intensity of the field. It should be mentioned here that the time behaviour of the atomic inversion and photon number has been investigated in ref. [ 221. The appearance of the so-called Cummings collapse, revivals and recorrelations has been shown for this multiphoton model. We introduce the two slowly varying hermitian quadrature components a I, u2 of the field, defined by a, =j[ue
i(wr-0)
_tu
u2=(1/2i)[aei(“‘-e)_u+e-i(wl-s)],
+e
-i(wr-0)
19
(7)
where 19is a phase angle that may be chosen at will. The commutation of a, and a2 is [u,, uz] = hi. The variances (Au,)~E (a f) - (u,)~ (i= 1, 2) satisfy the uncertainty relation (Au,)~(Au~)~> +e A state of the field is said to be squeezed when one of the quadrature components a, and a2 satisfies the relation (AU,)2<(AUi)$,h=i [18]. The condition for squeezing in the quadrature component can simply be written as S,
(8)
= -m(m-1)A+‘(gt)2,
if
m=l,
if
m>2,
(12)
These negative expressions indicate the immediate appearance of sueezing in a, for any photon multiple m and arbitrary nonzero intensity A after switching on the atom-field interaction. Such a behaviour is absent in the case when the atom is initially in the excited state [ 181. Analogously, the asymptotic expressions of S2 at very short times (gt << 1) are found from eqs. (11 b), (5) and (4) to be s, N fn(gt)4 , =m(m-
1)A*-‘(gt)2
,
if
m=l,
if
ma2.
(13)
These positive expressions indicatre the lack of squeezing in a2 at the beginning of interaction for any photon multiple m and any initial field intensity n. It should be noted that for the particular case m = 1 the first equation in (13) and the fact that squeezing occurs in a, at the onset of interaction, are in agreement with the results obtained recently by Butler and Drummond [ 201 for a cooperative Dicke system.
5 October 1987
PHYSICS LETTERS A
Volume 124, number 6,7
Fig. 1 presents the time evolution of S, computed numerically from eqs. (1 ), (5) and (4) for various intensities A of the coherent initial field and various photon multiples m. As soon as t>O, we observe nonclassical negative values of S,. For the cases (m=l,fi=0.2), (m=2,A=l.l2)and(m=3,A=3) the maximum magnitudes of squeezing in the region of short times are S, x - 0.28, - 0.49 and - 0.57 (i.e. 28%, 49% and 57%), respectively. As time goes on, S, starts oscillating and then reaches positive values. The long-time behaviour of S, is characterized by recoveries of squeezing. The squeezing in a, appears,
disappears, and later may appear again, see fig. Id. The maximum magnitude of squeezing recovered ~~1.12, m=2, see again (e.g., x52%forgt=l7.28, fig. 1d) may be larger than the maximum magnitude of squeezing in the short-time region ( ~49% for gtx0.92). It is seen from fig. 1 that for the larger intensity fi the duration of the first squeezing is generally shorter (except for the cases when overlapping of the first and second squeezing regions occurs). In figs. 2a and 2b, we plot S, versus time gt for the cases (m=l, E~0.2) and (m=2, E=5). It is clear from fig. 2 that squeezing may also occur in the field
0.1
0 cn’ -0.1
- 0.2 m=2
-0.6 0
0.4
-0.3 0
2
1
1.2
1.6
2.0
12
16
20
gt -
5
3
0.8
gt -
-0.6 _ 0
CC) _ m=3 0.2
0.L
0.6 qt -
0.8
1.0
1.2
0
4
8 gt -
Fig. 1. Time evolution of S,. (a) m = 1, short times: gf < 5. Here, the maximum magnitude S , z -0.28 of the first squeezing occurs for ti% 0.2 at gt = 2.75. (b) m = 2, short times gt < 2. The maximum magnitude 5’, z - 0.49 of the first squeezing occurs for ffz 1.12at gt= 0.92. (c) m= 3, short times gtc 1.2. The maximum magnitude S ,x -0.57 of the first squeezing occurs for ii=3 at gtz0.27. (d) Long times: gl< 20. The full line corresponds to the case m = 1,A= 0.2. The dashed line corresponds to the case m = 2, A= 1.12.The large magnitudes form=1.fi=0.2,gr~l965and.S of squeezing: S ,z-0.31 ,~-0.52form=2,ri=1.12,gt~17.28areobtained.
353
Volume
124, number
6.7
PHYSICS
(a) i
0.6 0.5 0.4
LETTERS
A
5 October
1987
able choice of the field intensity, the large magnitudes of squeezing have been obtained. It is however very difficult to detect this transient squeezing effect in an experiment. In recent experiments with Rydberg atoms in high-Q cavities it was possible to measure only the inversion of the atom but not the field itself. The connection between the evolution of the squeezing and that of the atomic inversion will be the subject of a subsequent paper.
References
[ I] R.E. Slusher, L.W. Hollberg, B. Yurke, J.C. Mertz and J.F. - 0.3 I 0
4
8
12
J 20
16 qt -
2.9
3.0
3.1
3.2
3.3
gt Fig. 2. Time evolution of&. (a) M= 1, A=0.2, long times: gt<20. (b) m=2, ri=S, the time interval 2.9cgfc3.3 for the first squeezing in uz.
component a2. The delay of squeezing in a2 for the cases examined here is seen. To conclude briefly, we have obtained squeezed states in the exactly solvable multiphoton Jaynes-Cummings model, where the atom is initially in the ground state and the fields is in a coherent state. The immediate appearance of squeezing in a, for short times of interaction has been established. Due to the photon multiplicity and the suit-
354
Valley, Phys. Rev. Lett. 55 (1985) 2409. [2] R.M. Shelby, M.D. Levenson, S.H. Perlmutter, R.G. Devoe and D.F. Walls, Phys. Rev. Lett. 57 (1986) 691. [ 31 Ling An Wu, H.J. Kimble, J.L. Hall and Huifa Wu, Phys. Rev. Lett. 57 (1986) 2520. [4] D.W. Walls, Nature 306 (1983) 141. [ 51 M.J. Collett and D.F. Walls, Phys. Rev. A 32 (1985) 2887. [6] J.R. Klauder, S.L. McCall and B. Yurke, Phys. Rev. A 33 (I 986) 3204. [ 71 L. Mandel, Opt. Commun. 42 (1982) 437. [8] M.H. Hillery, MS. Zubairy and K. Wodkiewicz, Phys. Lett. A 103 (1984) 259. [9] R. Loudon, Opt. Commun. 49 (1984) 67. [IO] H.P. Yuen, Phys. Rev. A 13 (1976) 2226. [ 1 I ] L.A. Lugiato and G. Strini, Opt. Commun. 41 (1982) 374. [ 121 M.D. Reid and D.F. Walls, Phys. Rev. A 28 (1983) 332. [ 131 D.F. Walls and P. Zoller, Phys. Rev. Lett. 47 (198 I ) 709. [ 141 P.A. Lakshmi and G.S. Agarwall, Phys. Rev. A 29 (1984) 2260. [ 151 S. Haroche and J.M. Raimond, in: Advances in atomic and molecular physics, Vol. 20, eds. D.R. Bates and B. Bederson (Academic Press, New York, 1985) p. 347. [ 161 G. Rempe, H. Walther and N. Klein, Phys. Rev. Lett. 58 (1987) 353. [ 171 H.I. Yoo and J.H. Eberly, Phys. Rep. 118 (1985) 239. [ IS] P. Meystre and M.S. Zubairy, Phys. Lett. A 89 (1982) 390. [ 191 G. Compagno, J.S. Peng and F. Persico, Opt. Commun. 57 (1986) 415. [20] M. Butler and P.D. Drummond, Opt. Acta 33 (1986) 1. [21] P.L. Knight, Phys. Ser. T 12 (1986) 51. [ 221 S. Singh, Phys. Rev. A 25 (1982) 3206.